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Properties

Label 1-6003-6003.1112-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.279 - 0.960i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.307 + 0.951i)2-s + (−0.810 + 0.585i)4-s + (−0.997 − 0.0679i)5-s + (−0.694 + 0.719i)7-s + (−0.806 − 0.591i)8-s + (−0.242 − 0.970i)10-s + (0.446 − 0.894i)11-s + (−0.128 + 0.991i)13-s + (−0.897 − 0.440i)14-s + (0.314 − 0.949i)16-s + (−0.540 − 0.841i)17-s + (−0.994 + 0.101i)19-s + (0.848 − 0.529i)20-s + (0.988 + 0.149i)22-s + (0.990 + 0.135i)25-s + (−0.983 + 0.182i)26-s + ⋯
L(s)  = 1  + (0.307 + 0.951i)2-s + (−0.810 + 0.585i)4-s + (−0.997 − 0.0679i)5-s + (−0.694 + 0.719i)7-s + (−0.806 − 0.591i)8-s + (−0.242 − 0.970i)10-s + (0.446 − 0.894i)11-s + (−0.128 + 0.991i)13-s + (−0.897 − 0.440i)14-s + (0.314 − 0.949i)16-s + (−0.540 − 0.841i)17-s + (−0.994 + 0.101i)19-s + (0.848 − 0.529i)20-s + (0.988 + 0.149i)22-s + (0.990 + 0.135i)25-s + (−0.983 + 0.182i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.279 - 0.960i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.279 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2173080498 + 0.2894966666i\)
\(L(\frac12)\) \(\approx\) \(-0.2173080498 + 0.2894966666i\)
\(L(1)\) \(\approx\) \(0.5926978934 + 0.4568166092i\)
\(L(1)\) \(\approx\) \(0.5926978934 + 0.4568166092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.307 + 0.951i)T \)
5 \( 1 + (-0.997 - 0.0679i)T \)
7 \( 1 + (-0.694 + 0.719i)T \)
11 \( 1 + (0.446 - 0.894i)T \)
13 \( 1 + (-0.128 + 0.991i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (0.709 + 0.704i)T \)
37 \( 1 + (0.359 + 0.933i)T \)
41 \( 1 + (0.371 + 0.928i)T \)
43 \( 1 + (-0.709 + 0.704i)T \)
47 \( 1 + (0.680 - 0.733i)T \)
53 \( 1 + (-0.0203 + 0.999i)T \)
59 \( 1 + (0.995 - 0.0950i)T \)
61 \( 1 + (0.885 + 0.464i)T \)
67 \( 1 + (-0.894 + 0.446i)T \)
71 \( 1 + (0.488 - 0.872i)T \)
73 \( 1 + (-0.122 + 0.992i)T \)
79 \( 1 + (0.746 + 0.665i)T \)
83 \( 1 + (-0.288 - 0.957i)T \)
89 \( 1 + (-0.396 - 0.917i)T \)
97 \( 1 + (-0.999 + 0.00679i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.4453484564077531893309670400, −16.70750485852480028955511372992, −15.737285430196278702323295256848, −15.09835052046886942258895308097, −14.76954358944926214430336139645, −13.813099646717401176559339748360, −13.04752199518069356932445663220, −12.591869944298069022448349831864, −12.164569702355174964013666145511, −11.18973802591566717696107674901, −10.72045388417858570091255466706, −10.14194909147791448167280942334, −9.46443242580634876745309171289, −8.60377300974018591834269717425, −7.99035696811045707470786234293, −7.09990656651535431416065927633, −6.4602984153771436185511646158, −5.574665938784567114401106377703, −4.5757473314092282447933445870, −4.04616459061502886453039353410, −3.65392577400017638980539989445, −2.71066871600589790834600082126, −1.98985676789187999011033350055, −0.85347009351388651988245257018, −0.12017263782640255611914333369, 1.01268379599371367730876575650, 2.53760271714101094934465572179, 3.15972508811980678151540839690, 3.98277375630832140172128794059, 4.52669298544270014366471963336, 5.281108443208586893497979019787, 6.29095070026770305720298484413, 6.59076558067947203179213438411, 7.27492896064656907121619045664, 8.27458348455002170366240544405, 8.65949205159959940705535949212, 9.20975610878790156597013908171, 10.00834746915486653376133813799, 11.17953884349431249938994465952, 11.78947453143631320357148623485, 12.208804716253804236822889496185, 13.09639795691282187316314162183, 13.58723353764180117828928443142, 14.458689591122796649714110422170, 14.95485217812371300686192164214, 15.67104239498094363388739533243, 16.17228014582975181732320732312, 16.59893067889781486890781184491, 17.24371707529139109500210383656, 18.332971044387016952998891418721

Graph of the $Z$-function along the critical line